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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>] <b>means </b><font color="Maroon">a<sub>1</sub></font> is <a href="group_4.html#V2" target="_self">maximal</a>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_1_1"/>
(  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>] or for <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>  holds not <font color="Olive">H</font> is <a href="group_4.html#V2" target="_self">maximal</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_1"/><i><font color="Green">E31</font></i>: 
 ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>]
 ;<br/><div class="add"><a NAME="E2:48_1_1_1_1_1"/>
 ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> :  ex <font color="Olive">K</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">K</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">K</font>] ) </span>}</span> 
 <b>from </b><i><a class="ref" href="group_4.html#S1" target="_self">GROUP_4:sch 1</a>(<a class="txt" href="group_4.html#E1:48_1_1_1_1_1"><i><font color="Green">E31</font></i></a>);<br/></i><b>hence </b><a NAME="E3:48_1_1_1_1_1"/>
( (  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <font color="Olive">H</font> is <a href="group_4.html#V2" target="_self">maximal</a> implies  ex <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> :  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H</font> &amp; <font color="Olive">H</font> is <a href="group_4.html#V2" target="_self">maximal</a> ) </span>}</span>  ) &amp; ( ( for <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>  holds not <font color="Olive">H</font> is <a href="group_4.html#V2" target="_self">maximal</a> ) implies  ex <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> #) ) )
 <b>by </b><i><a class="txt" href="group_4.html#E1:48_1_1_1_1_1"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>  holds not <font color="Olive">H</font> is <a href="group_4.html#V2" target="_self">maximal</a>
 ;<br/><div class="add"><a NAME="E2:48_1_1_1_1_2"/>
(  <a href="group_2.html#K6">(Omega).</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> #) &amp;  <a href="group_2.html#K6">(Omega).</a> <font color="Maroon">G</font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> )
 ;<br/><b>hence </b><a NAME="E3:48_1_1_1_1_2"/>
( (  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <font color="Olive">H</font> is <a href="group_4.html#V2" target="_self">maximal</a> implies  ex <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> :  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H</font> &amp; <font color="Olive">H</font> is <a href="group_4.html#V2" target="_self">maximal</a> ) </span>}</span>  ) &amp; ( ( for <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>  holds not <font color="Olive">H</font> is <a href="group_4.html#V2" target="_self">maximal</a> ) implies  ex <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> #) ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:48_1_1"/>
( (  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <font color="Olive">H</font> is <a href="group_4.html#V2" target="_self">maximal</a> implies  ex <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> :  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H</font> &amp; <font color="Olive">H</font> is <a href="group_4.html#V2" target="_self">maximal</a> ) </span>}</span>  ) &amp; ( ( for <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>  holds not <font color="Olive">H</font> is <a href="group_4.html#V2" target="_self">maximal</a> ) implies  ex <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> #) ) )
 ;<br/>


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